How to find the minimum of $x^2 + \frac{25}{x^2} + 3$ with proof based mathematics (no calculus)?

Question

I have a question where this is this expression:

$x^2 + \frac{25}{x^2} + 3$ where $x > 0$

And you have to find the minimum value of this expression. However, you cannot use calculus, such as graphing etc. but only certain proof related expression i.e. arithmetic-geometric mean inequality and basic algebra. I am not too sure what next step there is to it. Thanks.

(Note: I have answered by own question, if anyone can verify that it is true or not, please clarify and I will fix it or mark another solution as correct).

Answer 1

Okay, so I think the selected answer (I choose) is actually to an extent wrong.

This is because, the proof cannot be concluded with:

$x^2 + \frac{25}{x^2} +3 \geq 13$

You actually need to find a number in which the equation is true at $13$. Therefore, you cannot start with the steps of an expressions being $\geq 0$ because you wouldn't be able to conclude which numbers can actually "reach" the minimum of $13$. Therefore, you use the AGM (Arithmetic-Geometric mean) inequality, where $a = x^2$ and $b = \frac{25}{x^2}$ and solve accordingly. Then with that, you can find the appropriate numbers which satisfy the equation and conclude the proof correctly.

Answer 2

$$ \left( x - \frac{5}{x} \right)^2 \geq 0 $$ $$ x^2 - 10 + \frac{25}{x^2} \geq 0 $$ $$ x^2 + \frac{25}{x^2} \geq 10 $$ $$ x^2 + \frac{25}{x^2} +3 \geq 13 $$